Although ive only been once(last year) i would say look at Rhodes, a fantastic island and lots to see and do, we visited the acropolis at lindos and also did a land rover safari which took in quite a lot of the island and a very informative driver.
Im visiting Kos this year which also has a bit of good sight seeing to do but ive never been before so cant say much really
Im sure others will also give you some pointers too but wherever you decide im sure you will enjoy it

Another vote for Rhodes - as well as the acropolis at Lindos, you have Rhodes Town itself, built by the Knights of St John, and Petaloudes (Valley of the butterflies), the 7 springs, a couple of ancient city sites, and some great beaches.

Crete, you have the island of Spinalonga, it was a leper colony a book by Victoria Hislop called The Island will give you and insite to it, The palace of Knossos in Herakleon,, Chania , Souda Bay. there is so much to do there we went two years on the trot for two weeks and still didn't see everything,

Rhodes is beautiful, as said before

Kos Town is steeped in History again a beautiful place, you can go over to Nysyros a volcanic island .

Corfu also has a great deal of history, Corfu Town, then boat rides to Paxos and Anti Paxos again beautiful,

Something for you to think about there, but which every you choose I am sure it wont be the last time you go to the beautiful Greek Islands

Rhodes is ok for historic things to see, but a lot of the island has, as you say "souvenir shops and tourist traps"
Maybe look at the resort of Lardos on Rhodes. less "touristy" and a wonderful village square where you can sit during the day with a drink and watch the world go by.

For authentic Greek I would look at Lesvos (Lesbos) In my opinion, this is the least commersialised island I've been to, and I've been to a fair few. Take look at the resort of Petra here.

Crete is a good shout. Lots of historical places. But it is a very big island with 2 airports.

If it's picture postcard Greece you are after, then look at Santorini. There are 2 beach resorts, Kamari and Perissa. There are other non beach resorts, but these are on the "Caldera" and to stay here is very expensive. Depending on your budget, I'd choose one of the beach resorts.

Kos is another one. Lots of things to see and do, and great for cycling.

Parga on the mainland is good also.